Finite Group Databasebeta

← #496398

Subgroups — #496398

Order 512.

982 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#20723C211yesmaximal, characteristic
3#20723C211yesmaximal, characteristic
4#13299C214yesmaximal, complemented
5#13298C214yesmaximal, complemented
6#12859C214yesmaximal
7C22×C4⋊Q8C411yescharacteristic, complemented
8C2×C4×#41C2211yescharacteristic
9#2857—21no—
10#2857—21no—
11#2628—21no—
12#2628—21no—
13#2568C412yescomplemented
14#2567C412yescomplemented
15#2563C2212yes—
16#2542C2212yes—
17#2487—21no—
18#2487—21no—
19#2409—21no—
20#2409—21no—
21#2317C412yescomplemented
22#2311C2212yes—
23#2285C411yescharacteristic, complemented
24#2283—21no—
25#2283—21no—
26#1820—21no—
27#1820—21no—
28#1799—21no—
29#1799—21no—
30#1799C2212yes—
31#1799C2212yes—
32#1782C2212yes—
33#1782C2212yes—
34C4×#99—21no—
35C4×#99—21no—
36C4×#99C2212yes—
37C4×#99C2212yes—
38#1393C22116yes—
39C22×#41C2311yescharacteristic
40C23×Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—41no—
44C2×C4⋊Q8—41no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8C2×C412yes—
48C2×C4⋊Q8C2×C412yes—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C4×#41—41no—
52C4×#41—41no—
53C4×#41—41no—
54C4×#41—41no—
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×D4D411yescharacteristic
58C2×C4×D4D411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×C4×D4C2311yescharacteristic
62C2×C4×D4C2311yescharacteristic
63C2×#117C2×C411yescharacteristic
64C2×#117C2×C412yes—
65C2×#117D411yescharacteristic
66C2×#117D411yescharacteristic
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2×C412yes—
69C22×C42C2×C411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C2×#99—22no—
73C2×#99—22no—
74C2×#99C2314yes—
75C2×#99C2314yes—
76#400—22no—
77#400—22no—
78#399—22no—
79#399—22no—
80#395—22no—
81#395—22no—
82#393—22no—
83#393—22no—
84#391—22no—
85#391—22no—
86#391C2×C414yes—
87#390C2314yes—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#386—22no—
93#386—22no—
94#385—22no—
95#385—22no—
96#384—21no—
97#384C2×C414yes—
98#383—41no—
99#383—41no—
100#382—21no—
101#382—41no—
102#382—41no—
103#382C2×C414yes—
104#381—22no—
105#381—22no—
106#380—21no—
107#380—21no—
108#380—22no—
109#380—22no—
110#380—22no—
111#380—22no—
112#380—22no—
113#380—22no—
114C4×C4⋊C4—41no—
115C4×C4⋊C4—41no—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C412yes—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120#376—41no—
121#376—41no—
122#342—22no—
123#342—22no—
124#342—22no—
125#342—22no—
126C2×#41—24no—
127C2×#41—41no—
128C2×#41—41no—
129C2×#41—42no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8—84no—
135C22×Q8C22×C411yescharacteristic
136C22×D4C2×D411yescharacteristic
137C22×D4C2×D411yescharacteristic
138C22×D4C2411yescharacteristic
139C23×C4—22no—
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C4⋊Q8—42no—
150C4⋊Q8—42no—
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—41no—
156C4⋊Q8—41no—
157C4⋊Q8—41no—
158C4⋊Q8—41no—
159C22⋊Q8—84no—
160C22⋊Q8—84no—
161C4×Q8—41no—
162C4×Q8—41no—
163C4×Q8—41no—
164C4×Q8—41no—
165C4×D4—41no—
166C4×D4—41no—
167C4×D4—41no—
168C4×D4—41no—
169C4×D4—41no—
170C4×D4—41no—
171C4×D4—41no—
172C4×D4—41no—
173C4×D4—41no—
174C4×D4—41no—
175C4×D4—41no—
176C4×D4—41no—
177#117—41no—
178#117—41no—
179#117—41no—
180#117—41no—
181#117—41no—
182#117—41no—
183#117—44no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22×C414yes—
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4C22⋊C411yescharacteristic
211C2×C22⋊C4C22⋊C411yescharacteristic
212C2×C22⋊C4C2×D411yescharacteristic
213C2×C22⋊C4C2×D411yescharacteristic
214C2×C22⋊C4C2×D411yescharacteristic
215C2×C22⋊C4C2×D411yescharacteristic
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42C22⋊C411yescharacteristic
231C2×C42C22⋊C411yescharacteristic
232C2×C42C22⋊C411yescharacteristic
233C2×C42C22⋊C411yescharacteristic
234C2×C42C22⋊C411yescharacteristic
235C2×C42C22⋊C411yescharacteristic
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22×C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C414yes—
241#99—28no—
242#99—28no—
243#99—44no—
244#99—44no—
245#95—22no—
246#95—22no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—42no—
252#95—42no—
253#95—42no—
254#95—42no—
255#95—42no—
256#95—42no—
257#95—42no—
258#95—42no—
259C24C2×C22⋊C411yescharacteristic
260C24C22×D411yescharacteristic
261C24C22×D411yescharacteristic
262#41—48no—
263#41—48no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—44no—
268C2×Q8—44no—
269C2×Q8—44no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—84no—
273C2×Q8—88no—
274C2×Q8—88no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D42−1+414yes—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—84no—
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C23×C411yescharacteristic
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—84no—
348C4⋊C4—84no—
349C4⋊C4—84no—
350C4⋊C4—84no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—44no—
358C22⋊C4—44no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—44no—
362C22⋊C4—84no—
363C22⋊C4—84no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—44no—
383C42—84no—
384C42—84no—
385C42#9911yescharacteristic
386C42#9911yescharacteristic
387C42#9911yescharacteristic
388C42#9911yescharacteristic
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23#41012yes—
402C23C22×C22⋊C411yescharacteristic
403C23#53612yes—
404C23#53612yes—
405Q8—44no—
406Q8—84no—
407Q8—88no—
408Q8—84no—
409D4—44no—
410D4—44no—
411D4—44no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—88no—
461C2×C4—84no—
462C2×C4—84no—
463C2×C4—84no—
464C2×C4—84no—
465C2×C4—84no—
466C2×C4—84no—
467C2×C4C2×#9911yescharacteristic
468C2×C4C2×#9911yescharacteristic
469C2×C4C2×#9911yescharacteristic
470C2×C4C2×#9911yescharacteristic
471C2×C4C2×#9911yescharacteristic
472C2×C4C2×#9911yescharacteristic
473C2×C4#51912yes—
474C22—21no—
475C22—21no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—81no—
490C22—81no—
491C22#231011yescharacteristic
492C22C22×#9911yescharacteristic
493C22C2×#41011yescharacteristic
494C4—21no—
495C4—21no—
496C4—22no—
497C4—21no—
498C4—21no—
499C4—41no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—41no—
504C4—41no—
505C4—41no—
506C4—41no—
507C4—84no—
508C4—84no—
509C4—164no—
510C4—164no—
511C4#212812yes—
512C4#212912yes—
513C2—21no—
514C2—21no—
515C2—42no—
516C2—42no—
517C2—42no—
518C2#2182511yescharacteristic
519C2C2×#212811yescharacteristic
520C2C2×#212911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #13299C2semidirect
5. #13298C2semidirect
7. C22×C4⋊Q8C4semidirect
13. #2568C4semidirect
14. #2567C4semidirect
21. #2317C4semidirect
23. #2285C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#352792
#207232
#132992
#132982
#128592